Hadronic decays of eta and eta-prime with coupled channels

23 Pages·0.27 MB·English

by  N. Beisert

′
Hadronic decays of η and η
with coupled channels 1
N. Beisert2 and B. Borasoy3
3
0
0
Physik-Department, Technische Universit¨at Mu¨nchen
2
D-85747 Garching, Germany
n
a
J

9

1
v
8
5 Abstract
0
1
The hadronic decays η πππ, η πππ and η ηππ are investigated within
0 → ′ → ′ →
a U(3) chiral unitary approach. Final state interactions are included by deriving
3
0 the effective s-wave potentials for meson meson scattering from the chiral effective
/
h Lagrangian and iterating them in a Bethe-Salpeter equation. With only a small set
p of parameters we are able to explain both rates and spectral shapes of these decays.
-
p
e
h
:
v PACS: 12.39.Fe, 13.25.Jx
i
X
Keywords: Chiral effective lagrangians, hadronic decays of mesons,
r
a η and η′, coupled channels, final state interactions.
1Work supported in part by the DFG
2email: [email protected]
3email: [email protected]
1 Introduction
Thehadronicdecaysη πππ andη πππ areofgreatinterest since theyviolateisospin
′
→ →
symmetry. At such low energies the calculations of the decay amplitudes are based on
effective chiral Lagrangians which have the same symmetries and symmetry breaking
patterns as the underlying QCD Lagrangian. They are written in terms of effective
degrees of freedom, usually the octet of pseudoscalar mesons – pions, kaons and the eta –
[1], however, they can easily be extended to include the η also [2]. With the effective
′
Lagrangian at hand, one can perform a perturbative expansion of the decay amplitudes in
powers of the pseudoscalar meson masses and external momenta. For the decay η πππ
→
the expansion parameters are the ratios of the Goldstone boson octet masses or Lorentz
invariantcombinationsofexternalmomentaoverthescaleofspontaneouschiralsymmetry
breaking, m2P/Λ2χ and (pP ·pP′)/Λ2χ, respectively, with Λχ = 4πfπ ≈ 1.2 GeV and higher
loops contribute to higher chiral orders. At low energies these ratios are considerably
smaller than unity and the chiral expansion is expected to converge.
The inclusion of the η , on the other hand, spoils the conventional chiral counting
′
scheme, since its mass mη′ does not vanish in the chiral limit so that higher loops will still
contribute to lower chiral orders. This can be prevented by imposing large N counting
c
rules within the effective theory in addition to the chiral counting scheme. In the large
N limit the axial anomaly vanishes and the η converts into a Goldstone boson. The
c ′
properties of the theory may then be analyzed in a triple expansion in powers of small
momenta, quark masses and 1/Nc, see e.g. [2, 3, 4, 5]. In particular, mη′ is treated as a
small quantity. Phenomenologically, this is not the case since mη′ = 958 MeV, and we
will therefore treat the η as a massive state leading to a situation analogous to chiral
′
perturbation theory (ChPT) with baryons. Recently, a new regularization scheme – the
so-called infrared regularization – has been proposed which maintains Lorentz and chiral
invariance explicitly at all stages of the calculation while providing a systematic counting
scheme for the evaluation of the chiral loops [6]. Infrared regularization can be applied
in the presence of any massive state and has been employed in chiral perturbation theory
including the η (U(3) ChPT) in [7, 8, 9], but its applicability is restricted to very small
′
external three momenta. This is surely not the case for the decay η πππ where large
′
→
unitarity corrections are expected due to final state interactions between the three pions.
Final state interactions have already been shown to be substantial in η πππ both in
→
a complete one-loop calculation in SU(3) ChPT [10] and under the inclusion of final state
interactions using extended Khuri-Treiman equations [11]. The last calculation comes
closest to the experimental decay rate but still remains below it, if the usual quark mass
ratios are employed. However, the experimental rate of the decay can be reproduced
by increasing the quark mass ratio [(m m )/(m mˆ)] [(m + m )/(m + mˆ)] from
d u s d u s
− − ·
1/(24.1)2 to 1/(22.4 0.9)2 which isnow considered to bethe most accuratedetermination
±
of this quark mass ratio. It is therefore obligatory to include final state interactions also
in the decay η πππ. We will apply the following approach in the present investigation:
′
→
after deriving the effective potentials for meson meson scattering from the chiral effective
Lagrangian, we iterate them utilizing a Bethe-Salpeter equation (BSE). This method
has been proven to be useful both in the purely mesonic sector and under the inclusion
of baryons [12, 13]. The BSE generates dynamically bound states of the mesons and
baryons and accounts for the exchange of resonances without including them explicitly.
2
The usefulness of this approach lies in the fact that from a small set of parameters a large
variety of data can be explained. It has been extended recently to U(3) ChPT in [14]
where effects of the η in meson meson scattering are investigated and the parameters
′
of the U(3) chiral effective Lagrangian are constrained by comparing the results with
the experimental phase shifts. Again, with only a few chiral parameters the phase shifts
have been reproduced. Using the same technique, the electroproduction of the η meson
′
on nucleons has also been investigated yielding good agreement with data [15]. This
method allows us to account for final state interactions and to study the importance
of resonances in the different isospin channels, a topic of interest, e.g., in the dominant
hadronic decay mode of the η , η ηππ. A full one-loop calculation of this decay
′ ′
→
has been performed in [9] utilizing infrared regularization and reasonable agreement with
experimental data was achieved. However, both higher order final state interactions and
the explicit inclusion of resonances have been omitted. It has been claimed, on the
other hand, that this decay can be described in a tree-level model via the exchange of
the scalar mesons σ(560),f (980) and a (980) which are combined together with κ(900)
0 0
into a nonet [16] (see also [17, 18, 19, 20, 21]). The authors find that the exchange of
the scalar resonance a (980) dominates. By iterating the effective chiral potentials to
0
infinte order in a Bethe-Salpeter equation, we can investigate the importance of these
resonances explicitly. Hence, the present work provides a unified approach to the decays
η πππ, η πππ and η ηππ with chiral symmetry and unitarity being the main
′ ′
→ → →
ingredients. With only a few chiral parameters we can compare our results with a variety
of experimental data and even predict the decay rate for η π0π+π . For simplicity, we
′ −
→
will restrict ourselves to s-waves since good agreement with experimental data is already
obtained; an extension to higher multipoles is straightforward.
In the next section, we present the chiral effective Lagrangian up to fourth chiral order
and the π0ηη mixing for different up- and down-quark masses m = m is discussed in
′ u d
6
detail as it arises from the second and fourth order Lagrangian, respectively. In Section 3
the implementation of the final state interactions via the BSE is illustrated. In Sections 4
to 4.1 the decays η πππ, η πππ and η ηππ are discussed. Our results are
′ ′
→ → →
compared with experimental data.
2 The Lagrangian density
In this section the Lagrangians at second and fourth chiral order are presented and the
resulting π0-η -η mixing is discussed. The effective Lagrangian for the pseudoscalar
8 0
meson nonet (π,K,η ,η ) reads up to second order in the derivative expansion [2, 5, 7]
8 0
(0+2) = V +V ∂ U ∂µU +V U χ+χ U +iV U χ χ U +V U ∂µU U ∂ U , (1)
0 1 µ † 2 † † 3 † † 4 † † µ
L − h i h i h − i h ih i
whereU isaunitary3 3matrixcontainingtheGoldstonebosonoctet(π˜ ,π˜0,K˜ ,K˜0,η )
± ± 8
×
and the η . Its dependence on π˜0,η and η is given by
0 8 0
U = exp diag(1, 1,0) iπ˜0/F +diag(1,1, 2) iη /√3F +i√2η /√3F +... . (2)
8 0
− · − ·
(cid:0) (cid:1)
The expression ... denotes the trace in flavor space, F is the pion decay constant in the
h i
chiral limit and the quark mass matrix = diag(m ,m ,m ) enters in the combination
u d s
M
3
χ = 2B with B = 0 q¯q 0 /F2 being the order parameter of the spontaneous sym-
M −h | | i
metry violation. As we do not consider external (axial-) vector currents in the present
investigation, the covariant derivatives have been replaced by partial ones.
The coefficients V are functions of η , V (η /F), and can be expanded in terms of this
i 0 i 0
variable. At a given order of derivatives of the meson fields U and insertions of the quark
mass matrix one obtains an infinite string of increasing powers of η with couplings
0
M
which are not fixed by chiral symmetry.4 Parity conservation implies that the V are all
i
even functions of η except V , which is odd, and V (0) = V (0) = V (0) 3V (0) = 1F2
0 3 1 2 1 − 4 4
gives the correct normalizaton for the quadratic terms of the mesons. The potentials V
i
are expanded in the singlet field η
0
η η2 η4
V 0 = v(0) +v(2) 0 +v(4) 0 +... for i = 0,1,2,4
i F i i F2 i F4
hη i η η3
V 0 = v(1) 0 +v(3) 0 +... (3)
3 F 3 F 3 F3
h i
(j)
with expansion coefficients v to be determined phenomenologically.
i
In order to describe the isospin violating decays η πππ and η πππ, we need to
′
→ →
distinguish between the up- and down-quark masses, m and m , which leads to π˜0-η -η
u d 8 0
mixing. Taking the Lagrangian from Eq. (1), diagonalization of the mass matrix to first
order in isospin breaking m m yields the mass eigenstates π0,η,η with
u d ′
−
π˜0 = π0 ǫη 2ǫϑη
′
− −
η = ǫπ0 +η ϑη
8 ′
−
η = 3ǫϑπ0 +ϑη +η . (4)
0 ′
The angles ǫ and ϑ arise from the mass differences of the light quark masses m ,m ,m .
u d s
While ϑ breaks SU(3) symmetry due to different strange and nonstrange quark masses,
ǫ is proportional to strong isospin violation m m
d u
−
√3m m m2
ǫ= d − u = ǫ
4 m mˆ √3(m2 m2)
s − η − π
4√2v˜(1) √2v˜(1)
ϑ= 2 B(m mˆ) = 2 (m2 m2) (5)
3v(2) s − v(2) η − π
0 0
where we used the abbreviations
v˜(1) = 1F2 1√6v(1)
2 4 − 2 3
m2 = B(m m )
ǫ d − u
mˆ = 1(m +m ). (6)
2 d u
Thequantity m2 canbeexpressed intermsofphysical mesonmasses byapplying Dashen’s
ǫ
theorem [22], which implies the identity of the pion and kaon electromagnetic mass shifts
up to (e2p2)
O
m2 = m2 m2 +m2 m2 . (7)
ǫ K0 − K± π± − π0
4Note that we do not make use of 1/N counting rules.
c
4
At leading order in the chiral expansion the meson masses are given by
m2 =2Bmˆ,
π
m2 =B(m +m ),
K± s u
m2 =B(m +m ). (8)
K0 s d
Isospin violation is known to be small, hence ǫ is a small quantity despite being of zeroth
order in the quark masses. Terms of order (m m )2 are tiny and therefore neglected
d u
−
throughout. In the isospin limit of equal up- and down-quark masses m = m the angle
u d
ǫ vanishes and π˜0 does not undergo mixing with η and η .
8 0
However, this is not the whole story. As shown in [8] the fourth order Lagrangian
contributes to η-η mixing at leading order as well which is due to the fact that we count
′
the η mass as quantity of zeroth chiral order. The fourth order Lagrangian has the form
′
(4) = β (η )O (9)
L i i 0 i
with the fourth order operators P
O = CµC M , O = CµC M ,
4 µ 5 µ
−h ih i −h i
O = Cµ C M , O = Cµ C M ,
17 h ih µih i 18 −h ih µ i (10)
O = M M , O = N N ,
6 7
h ih i h ih i
O = 1 MM +NN , O = 1 MM NN ,
8 2h i 12 4h − i
We used the abbreviations C = U ∂ U, M = U χ + χ U, N = U χ χ U and the
µ † µ † † † †
−
coefficients β can be expanded in η in the same manner as the v in (3). The mixing
i 0 i
between π˜0,η and η cannot be described in terms of just two angles any longer. Due to
8 0
the operators in Eq. (10) two subsequent transformations of the original fields π˜0,η and
8
η are necessary to bring both the kinetic and mass terms of the Lagrangian into diagonal
0
form. At leading order in isospin breaking the new transformed fields are then related to
the original ones by
π˜0 = (1+Rπ˜0π0)π0 +Rπ˜0ηη +Rπ˜0η′η′
η8 = R8π0π0 +(1+R8η)η+R8η′η′
η0 = R0π0π0 +R0ηη +(1+R0η′)η′ (11)
with the mixing parameters given by
m2
R(0) = ǫ , R(0) = R(0) ,
8π0 √3(m2 m2) π˜0η − 8π0
η − π
R(2) = R(0) (R(2) + 2∆ ), R(2) = R(0) (R(2) + 2∆ ),
8π0 8π0 π˜0π0 3 GMO π˜0η − 8π0 8η 3 GMO
4v˜(1)(m2 m2) 8β(0) (m2 m2)
R(2) = 2 η − π , R(2) = R(2) + 5,18 η − π ,
0η √2F2m2 8η′ − 0η √2F2
0 (12)
(2) (0) (2) (2) (0) (2)
R = 3R R , R = 2R R ,
0π0 8π0 0η π˜0η′ 8π0 8η′
4β(0)m2 +6β(0)(m2 +m2) 2β(0) (m2 +m2)
(2) 5 π 4 η π (2) 4,5,17,18 η π
R = , R = ,
π˜0π0 − F2 0η′ − F2
4β(0)m2 +6β(0)(m2 +m2)
(2) 5 η 4 η π
R = ,
8η − F2
5
where the superscript denotes the chiral order and we have employed the abbreviations
β(0) = 3β(0)+β(0) 9β(0)+3β(0) andβ(0) = β(0)+3β(0)/2. Thequantitym2 = 2v(2)/F2
4,5,17,18 4 5 − 17 18 5,18 5 18 0 0
is the mass of the η meson in the chiral limit and the deviation from the Gell-Mann–
′
Okubo mass relation for the pseudoscalar mesons is given by
4m2 m2 3m2 6(m2 m2) 4(v˜(1))2
∆ = K − π − η = η − π β(0) 6β(0) 12β(0) + 2 . (13)
GMO m2 m2 F2 5 − 8 − 7 F2m2
η − π (cid:20) 0 (cid:21)
Finally, the remaining terms of the fourth order Lagrangian that we need in this
calculation are given by
O = CµCνC C , O = CµC CνC ,
0 µ ν 1 µ ν
h i h ih i
O = CµCν C C , O = CµC CνC ,
2 µ ν 3 µ ν
h ih i h i
O = Cµ C CνC , O = Cµ C CνC ,
13 µ ν 14 µ ν
−h ih i −h ih ih i
O = Cµ Cν C C , O = Cµ C Cν C , (14)
15 µ ν 16 µ ν
−h ih ih i h ih ih ih i
O = CµC iN , O = CµC iN ,
21 µ 22 µ
h i h ih i
O = Cµ C iN , O = Cµ C iN ,
23 µ 24 µ
h ih i h ih ih i
O = iMN , O = M iN .
25 26
h i h ih i
Usually the β term is not presented in conventional ChPT, since there is a Cayley-
0
Hamilton matrix identity that enables one to remove this term leading to modified co-
efficients β , i = 1,2,3,13,14,15,16 [4], but for our purposes it turns out to be more
i
convenient to include it. Hence, we do not make use of the Cayley-Hamilton identity
and keep all couplings in order to present the most general expressions in terms of these
parameters. One can then drop one of the β involved in the Cayley-Hamilton identity at
i
any stage of the calculation.
2.1 Values of LECs
The unknown coupling constants of the chiral Lagrangian – the so-called low-energy con-
stants (LECs) – need to be fit to experimental data. This has already been accomplished
in [14], where after applying the same approach as in the present investigation we con-
strained the LECs of the Lagrangian up to fourth chiral order by comparing the results
with the experimental phase shifts of meson meson scattering. Agreement was achieved
in the isospin I = 0, 1 channels up to 1.2 GeV and in the isopin I = 3,2 channels up
2 2
to 1.5 GeV. The discrepancey to the data above 1.2 GeV in the I = 0 channel, e.g., is
due to the omission of the 4π channel, see [23]. In order to obtain reasonable agreement
with the experimental phase shifts, the values of the involved chiral parameters are then
constrained to the following ranges
β(0) = (0.6 0.1) 10 3, β(0) = ( 0.5 0.1) 10 3,
0 ± × − 3 − ± × − (15)
β(0) = (1.4 0.2) 10 3, β(0) = (0.2 0.2) 10 3.
5 ± × − 8 ± × −
while keeping v(0) = v(0) = 1F2 and all remaining parameters set to zero. However,
1 2 4
variations of some of the parameters which have been set to zero do not yield any sig-
nificant effect for the phase shifts, and in principle they can have finite values. Such a
(0) (1) (0)
parameter is β or the combination v˜ from Eq. (6), and we have chosen β = 0 and
6 2 6
6
π0
η
s=(pη pπ0)2 π−
−
π+
Figure 1: Shown is a possible contribution to final state interactions in the decay
η π0π+π . Here, it takes place in the s-channel between π+π and the bubble chain
− −
→
represents part of the solution to the BSE at energy √s.
v˜(1) = v(0) = 1F2 in [14] but we are free to change their values in the present investi-
2 2 4
gation as long as they do not contradict the data for the phase shifts. As a matter of
fact, this work provides a test whether the same values for the LECs can be employed
when describing the hadronic decays of the η and η and will allow for finetuning of the
′
(0) (1)
parameters that were not constrained in [14], e.g. β and v˜ . A good fit to the decays
6 2
discussed in this work is given by
(1) (2)
v˜ = v˜ = 0,
2 2
β(0) = 0.56 10 3, β(0) = 0.3 10 3, (16)
0 × − 3 − × −
β(0) = 1.4 10 3, β(0) = 0.06 10 3.
5 × − 6 × −
with v˜(2) = 1F2 √6v(1) 3v(2) and all the remaining parameters being zero. Note
2 4 − 3 − 2
that there have been moderate changes with respect to the values in Eq. (15) which were
necessary to obtain better agreement with the experimental decay rates and the spectral
shapes of the Dalitz distributions. It is surprising that with a small number of parameters
we are able to explain a variety of data within the approach.
It is also important to note, that the parameter choice in Eq. (16) is not unique, since
variations in one of the parameters may be compensated by the other ones. Nevertheless,
we prefer to work with this choice; it is capable of describing the experimental phase
shifts and the hadronic decays of the η and η – as we will see shortly – with a minimal
′
set of parameters and motivated by the assumption that most of the OZI-violating and
(0) (1) (2) (1) (2)
1/N -suppressed parameters are not important, although β and v ,v in v˜ ,v˜
c 6 3 2 2 2
have small but non-vanishing values.
3 Final state interactions
An investigation of the hadronic decay η πππ at leading order in ChPT [24, 25]
→
yields a decay width which is significantly below the measured value [26]. The result is
improved by performing anext-to-leading order calculation, but stillfailsto reproduce the
phenomenological value. The substantial increase from leading to next-to-leading order
already indicates that higher order effects beyond one-loop ChPT may be important as
well and should not be neglected. Kambor, Wiesendanger and Wyler [11] were able to
approximate these higher order effects by using Khuri-Treiman equations, which describe
7
= + + +...
Figure 2: Solution to the Bethe-Salpeter equation T = A+AGA+AGAGA+...
final state interactions. The underlying idea is that the initial particle, i.e. the η or η ,
′
decays via chirally constrained vertices derived from the effective Lagrangian into three
mesons and that two out of these three mesons rescatter an arbitrary number of times,
see Fig. 1 for illustration. There are three possible ways in combining two of the mesons
to a pair while leaving the third one unaffected which corresponds to the s-, t- and u-
channel, respectively. Interactions of the third meson with the pair of rescattering mesons
are neglected. Such an infinite meson meson rescattering can alternatively be generated
by application of the Bethe-Salpeter equation. To this end, the s-wave potentials for
meson meson scattering are derived from the chiral effective Lagrangian and iterated in
a Bethe-Salpeter equation. The BSE then generates the propagator for two interacting
particles in a covariant fashion.
The Bethe-Salpeter equationfor thetwo-particle propagatorT fromalocalinteraction
A is given by [14, 27]
iddk T(p,q ,k)A(p,k,q )
i f
T(p,q ,q ) = A(p,q ,q )+ . (17)
i f i f (2π)d (k2 m2)(k¯2 m¯2)
Z − −
Combinatorial factors for two identical particles in the loophave to be taken into account,
but we prefer to keep this form of the BSE and modify the amplitudes accordingly. In
that case A must be the four-point amplitude from ChPT multiplied by 1 in order for
−2
T to be proportional to a bubble chain with the correct factors from perturbation theory.
The factor of 1 is the symmetry factor of two identical particle multiplets in a loop and
2
1 stems from factors of i in the vertices and propagators.
−
The solution for T amounts to an infinite resummation of the interaction kernels A
andthe free propagatorsG and can bediagrammaticaly depicted asan infinite interaction
chain, Fig. 2. In a short notation, both the equation and its solution can be written as
T = A+TGA, T = (1 GA) 1A = A+AGA+AGAGA+... (18)
−
−
For the decays considered here we would like to use these final-state interactions in such
a way as to approximate to a large extent the one-loop result from conventional ChPT.
The second term in the expansion of T, AGA, equals the unitary corrections in one-
loop ChPT (i.e. without tadpoles). Tadpoles which are not included in our approach
have been shown to yield numerically small effects in scattering processes in the physical
region and can furthermore be partially absorbed by redefining chiral parameters [12].
We have therefore neglected the tadpoles and crossed diagrams and restrict ourselves to
the tree diagrams from next-to-leading order ChPT in the calculation of the potential A.
As mentioned above, for the decays one must take into account the three possible ways
in combining two of the light mesons to a pair while leaving the third one unaffected,
which corresponds to the s-, t- and u-channel. Final state interactions will occur in all
of these channels and in order to obtain the full amplitude one must add an interaction
8
chain in each channel, T +T +T . This reproduces the unitary corrections at one-loop
s t u
order; the leading contribution in this sum, however, does not yield the tree level result
A from ChPT. Portions of the tree level amplitude are included in each T and must be
subtracted from the bubble chain. The proper expression can be easily accounted for by
choosing the full amplitude as T = A+(T A )+(T A )+(T A ) which includes
s s t t u u
− − −
the unitary corrections in one-loop ChPT, while the leading contact term pieces A are
s,t,u
removed from the solutions T and replaced by the tree diagram amplitude A.
s,t,u
In the evaluation of the BSE we will restrict ourselves to s-waves, since they are
expected to dominate in the decays discussed here, and indeed this simplification gives
results already in agreement with experiment. An improvement of the results – particu-
larly for the spectral shape of the Dalitz distributions – may be expected by including the
p-waves, but this is beyond the scope of the present investigation. We can further simplify
the integral in the BSE (17), since we are only interested in the physically relevant piece
of the solution T with all momenta put on the mass shell. The amplitude A contains in
general off-shell parts which deliver via the integral a contribution even to the on-shell
part of the solution T. However, these off-shell parts yield exclusively chiral logarithms
which – besides being numerically small – can be absorbed by redefining the regulariza-
tion scale of the loop integral. Furthermore the off-shell parts are not uniquely defined
in ChPT. We will therefore set all the momenta in the amplitudes in (17) on-shell, which
has also been done in previous work, see e.g. [12]. Finally, we need to consider all two-
meson channels. In particular, those which have the same set of quantum numbers can
couple amongst each other. Consequently, the objects A,T,G are promoted to matrices
in channel space and are functions of the two particle invariant mass squared p2. With
these simplifications the BSE equation reduces to (18) as a matrix equation for every p2
and the solution is obtained by matrix inversion. The matrix G is a diagonal matrix with
elements given by the integral
ddk i
G (p2) = (19)
mm¯ (2π)d (k2 m2)((k p)2 m¯2)
Z − − −
where m and m¯ are the masses of the two mesons in the corresponding channel. In
dimensional regularization the integral is evaluated as
1 mm¯ m2 m¯2 m
G (p2) = 1+ln + − ln
mm¯ 16π2 − µ2 p2 m¯
(cid:20)
2 λ (p2) λ (p2)
mm¯ mm¯
artanh
− p2 (m+m¯)2 p2
p p − (cid:21)
λ (p2) = (m m¯)2 p2 (m+m¯)2 p2 . (20)
mm¯
− − −
(cid:0) (cid:1)(cid:0) (cid:1)
The integral is divergent and µ takes the role of a regularization constant which can be
chosen independently for every channel. With these simplifications we write the full four-
point amplitude T as a sum of the solutions T of the BSE in Eq. (17) in the s-, t-
abcd ab,cd
and u-channel
T (s,t,u) = A (s,t,u)+(T A) (s)+(T A) (t)+(T A) (u), (21)
abcd abcd ab,cd ac,bd ad,bc
− − −
9
where, e.g., (T A) (s) is a bubble chain in the s-channel with particles ab on one
ab,cd
−
side, particles cd on the other side, and the tree level piece removed. The expression
(T A) (s) is thus equivalent to the s-wave projection of the final-state interactions
ab,cd
−
of T, whereas the tree-level amplitude A (s,t,u) is not decomposed into partial waves.
abcd
In the considered decays, η,η πππ and η ηππ, the two-particle states are either
′ ′
→ →
uncharged or simply charged. There are nine uncharged combinations of mesons
π0π0,π+π ,ηπ0,ηη,K0K¯0,K+K ,η π0,η η,η η (22)
− − ′ ′ ′ ′
which are labeled with indices 1,...,9, respectively, and a set four charged channels
π0π+,ηπ+,K+K¯0,η π+ (23)
′
labeled with indices 1,...,4. Due to isospin breaking, couplings between the different
channels of a set occur, but charge conservation prevents transitions between both sets.
One obtains a 9 9 matrix A0 which summarizes the amplitudes for the uncharged
×
channels and a 4 4 matrix A+ for the charged channels. The division of the two-particle
×
states into charged and uncharged channels simplifies the treatment of the BSE. We have
presented the leading order contributions for A0 and A+ in the appendix.
In order to obtain the final expressions T for the decay amplitudes, the solutions
abcd
T0 and T+ of the BSE in Eq. (17) must be corrected due to the overall symmetry factor
1/2 which was initially absorbed into the matrix A. This is achieved by setting
−
Tη0,00 = −√2T301, Tη′0,00 = −√2T701,
TTηη0+,+,0+− == −−TT320+21,, TTηη′′0+,+,0+− == −−TT740+21,, (24)
Tη′η,00 = −√2T801, Tη′0,η0 = −T803,
Tη′η,+− = −T802, Tη′+,η+ = −T4+2,
where an index 0, denotes π0,π . The same relations hold between the amplitudes
±
A and A+,0. Th±e additional factors √2 in some of the channels are included in order
ab,cd ij
to account for the statistical factor occuring in states with identical particles (π0,π0).
4 η πππ
→
The decay η πππ violates isospin and can take only place due to a finite mass differ-
→
ence m m or electromagnetic interactions. The latter ones are expected to be small
u d
−
(Sutherland’s theorem) [28] which has been confirmed in an effective Lagrangian frame-
work [11]. Disregarding these, the amplitude is proportional to m m and provides a
u d
−
sensitive probe on SU(2) symmetry breaking.
The process η πππ is the dominant decay mode of the η with the measured rates
→
being [26]
Γ(η π0π0π0) = (379 40)eV
→ ±
Γ(η π0π+π ) = (274 33)eV (25)
−
→ ±
10